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%I #33 Sep 08 2022 08:45:14
%S 1,325,105949,34539049,11259624025,3670602893101,1196605283526901,
%T 390089651826876625,127168029890278252849,41456387654578883552149,
%U 13514655207362825759747725,4405736141212626618794206201,1436256467380108914901151473801,468215202629774293631156586252925
%N Pell equation solutions (9*b(n))^2 - 82*a(n)^2 = -1 with b(n):=A097738(n), n >= 0.
%H Indranil Ghosh, <a href="/A097739/b097739.txt">Table of n, a(n) for n = 0..397</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (326,-1).
%F a(n) = S(n, 2*163) - S(n-1, 2*163) = T(2*n+1, sqrt(82))/sqrt(82), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
%F a(n) = ((-1)^n)*S(2*n, 18*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
%F G.f.: (1-x)/(1- 326*x+x^2).
%F a(n) = 326*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=325. - _Philippe Deléham_, Nov 18 2008
%e (x,y) = (9*1=9;1), (2943=9*327;325), (959409=9*106601;105949), ... give the positive integer solutions to x^2 - 82*y^2 =-1.
%t LinearRecurrence[{326, -1},{1, 325},12] (* _Ray Chandler_, Aug 12 2015 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-326*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) I:=[1,325]; [n le 2 select I[n] else 326*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-326*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,325];; for n in [3..20] do a[n]:=326*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y Cf. A097737 for S(n, 326).
%Y Row 9 of array A188647.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
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